Method of creating and evaluating bandlimited noise for computer graphics

ABSTRACT

Function spaces defined by scaling functions are used to generate bandlimited noise octaves and other attribute data sets. Scaling functions are basis functions that admit multiresolution analysis and include piecewise constant scaling functions, piecewise polynomial scaling functions, bandlimited scaling functions, Daubeschies scaling functions, as well as other multiresolution analysis scaling basis functions known to those of skill in the art. Scaling basis functions can be locally supported or have infinite support. The properties of the scaling basis functions used to construct bandlimited noise octaves may ensure that any bandlimited noise octave at resolution level N is orthogonal to bandlimited noise octaves and their associated scaling basis functions at all resolution levels less than N. Bandlimited noise octaves can be scaled to any resolution level and guaranteed to have no effect on images at any lower resolution level.

CROSS-REFERENCES TO RELATED APPLICATIONS

This application claims priority to U.S. Provisional Patent ApplicationsNos. 60/647,726, filed Jan. 26, 2005, and 60/675,932, filed Apr. 21,2005, the disclosures of which are incorporated by reference herein forall purposes.

BACKGROUND OF THE INVENTION

The present invention relates to the field of computer graphics, and inparticular to methods and apparatus for animating computer generatedcharacters. The present invention relates to the field of computergraphics. Many computer graphic images are created by mathematicallymodeling the interaction of light with a three dimensional scene from agiven viewpoint. This process, called rendering, generates atwo-dimensional image of the scene from the given viewpoint, and isanalogous to taking a photograph of a real-world scene. Animatedsequences can be created by rendering a sequence of images of a scene asthe scene is gradually changed over time. A great deal of effort hasbeen devoted to making realistic looking rendered images and animations.

One important aspect in creating realistic looking rendered images isthe creation of attributes for the surfaces and optionally the interiorsof objects. Optical attributes can include color; transparency;reflective and refractive properties (such as diffuse and specularreflective coefficients; and geometric properties (such as surfacenormal and displacement vectors). Attributes often vary in complex waysover the surface or space defined by an object. To model these changes,one or more data sets defining all or a portion of an object'sattributes are defined and mapped onto the surfaces and/or volume of anobject. Data sets may have one, two, three, or more dimensions. Forexample, a texture map is a data set defining the color of the surfacesand/or volume of an object. Texture maps can be used to give objects theappearance of different materials. For example, a wood texture or amarble texture can be applied to an object to give it the appearance ofbeing made of wood or marble, respectively.

Attribute data sets can be generated in a number of ways. An artist canmanually create data sets such as texture maps by drawing an image ofthe desired data set. Other software tools can be used to manuallydefine other types of attribute data sets, such as surface normal ordisplacement vectors. Additionally, data sets can be generatedprocedurally. Typically, procedural data sets are created by applyingone or more algorithms to one or more input data sets. In someapplications, only the algorithms and input data sets for a proceduraldata set are defined in advance of rendering. During rendering, theprocedural data is produced as needed for only the portions of objectsbeing rendered. Procedural data sets can be used to approximate theattributes of a variety of different materials, such as wood, stone,glass, smoke, fire, and fur.

One type of input data often used by procedural data sets is noise.Noise is a set of random, pseudorandom, or other type of data withoutany discernable pattern. Data having a discernable pattern may also havea noise component, which is the portion of the data without adiscernable pattern. Noise is used by procedural data sets to introducenatural variations in the attributes of an object and to ensure that nounnatural repeating patterns occur. One common type of noise used byprocedural data sets is Perlin noise. Perlin noise creates a noise setby combining two or more octaves of random or pseudorandomly generateddata. An octave is a set of data having a frequency spectrum containedbetween two frequencies that have a two to one ratio. By combiningoctaves of data, Perlin noise attempts to bandlimit the resulting noiseto a desired frequency spectrum.

Aliasing due to undersampling high frequency data is one problem thatarises during rendering. Rendering is analogous to sampling the data ofa scene, with each pixel or sub-pixel representing a sample. Thedistance between pixels determines the maximum spatial bandwidth for agiven rendering resolution. If the scene contains data, includingattribute data sets, having high frequency components exceeding themaximum spatial bandwidth of the renderer, aliasing can occur.

One problem with Perlin noise is that it is poorly bandlimited. AlthoughPerlin noise is created from separate noise octaves, each octave oftenincludes frequency components outside the desired range of frequenciesof the octave. As a result, rendering procedural data sets created fromPerlin noise can result in portions that have aliasing artifacts due toundersampling of excess high frequency components or blurring due toexcess low frequency components. For scenes with surfaces sampled at avariety of different frequency ranges, for example a ground surfaceextending from the foreground to the horizon, the user must often choosebetween including additional octaves and introducing aliasing as aresult, or excluding these additional octaves and introducing blurringor loss of detail.

It is therefore desirable for a system and method to create bandlimitednoise data for procedural data sets and other purposes. It is furtherdesirable to provide greater control over the distribution andappearance of combinations of noise octaves. It is also desirable to beable to seamlessly combine attribute data sets having any arbitrarysizes and frequency spectrums and to efficiently filter the combinedresults to prevent aliasing, blurring, and other unwanted visualartifacts.

BRIEF SUMMARY OF THE INVENTION

An embodiment of the invention uses function spaces generated by scalingfunctions to generate bandlimited noise octaves and other attribute datasets. Scaling functions are basis functions that admit multiresolutionanalysis and include piecewise constant scaling functions, such as Haarbasis functions; piecewise polynomial scaling functions, such as uniformB-spline basis functions of any degree; bandlimited scaling functions,such as sinc functions; Daubeschies scaling functions, as well as othermultiresolution analysis scaling basis functions known to those of skillin the art. Scaling basis functions can be locally supported, allowingfor noise data to be constructed using a small number of noisecoefficients, or have infinite support.

In an embodiment, the properties of the scaling basis functions used toconstruct bandlimited noise octaves may ensure that any bandlimitednoise octave at resolution level N is orthogonal to bandlimited noiseoctaves and their associated scaling basis functions at all resolutionlevels less than N, such as resolution levels N-1, N-2, and so forth.This is true regardless of whether the scaling basis functions atdifferent resolution levels are orthogonal to each other, such as Haarscaling basis functions, or not orthogonal to each other, such asB-spline scaling basis functions. Moreover, an embodiment of thebandlimited noise octaves can be scaled to any resolution level andguaranteed to have no effect on images at any lower resolution level.

In an embodiment, a method of creating a bandlimited data set for use increating at least one computer graphics image includes receiving anunfiltered data set. The unfiltered data set has a first frequencyspectrum that includes a first maximum frequency. An embodiment of themethod transforming the unfiltered data set into a filtered data set.The filtered data set has a second frequency spectrum. The secondfrequency spectrum includes a second maximum frequency that is less thanthe first maximum frequency. An embodiment of the method processes thefiltered data set and the unfiltered data set to define a firstbandlimited data set having a frequency spectrum substantially betweenthe first maximum frequency and the second maximum frequency andcombines the first bandlimited data set with a second bandlimited dataset having a frequency spectrum substantially between a third frequencyand a fourth frequency to form the data set.

In an embodiment, the first bandlimited data set is defined bycoefficients of a set of locally supported scaling basis functions. Theset of locally supported scaling basis functions may include a piecewisepolynomial scaling basis function or a piecewise constant scaling basisfunction.

In an embodiment, processing the filtered data set and the unfiltereddata set includes converting the filtered data set and the unfiltereddata set to a common resolution and subtracting each element of theconverted filtered data set from its corresponding element in theconverted unfiltered data set. In a further embodiment, converting thefiltered data set and unfiltered data set includes upsampling thefiltered data set to a resolution of the unfiltered data set.

In another embodiment, transforming the unfiltered data set into thefiltered data set includes transforming the unfiltered data set todetermine detail coefficients that represent a portion of the unfiltereddata set that cannot be represented by the filtered data set. In thisembodiment, processing the filtered data set and the unfiltered data setto define a first bandlimited data set includes inverse transforming ofthe detail coefficients.

In an embodiment, receiving an unfiltered data set comprises receivingan attribute data set associated with a visible characteristic of anentity to be used in a computer graphics system. In another embodiment,the unfiltered data set is noise data lacking a discernable pattern,such as stochastically generated, randomly generated, or pseudo-randomlygenerated noise.

In a further embodiment, a sum of the frequency spectrum of the firstbandlimited data set and a copy of the frequency spectrum of the firstbandlimited data is substantially flat if the copy of the frequencyspectrum is translated in frequency by the difference between the firstand second maximum frequencies.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention will be described with reference to the drawings, inwhich:

FIG. 1 illustrates a block diagram of a computer system suitable forimplementing an embodiment of the invention;

FIGS. 2A-B illustrate a prior type of noise;

FIGS. 3A-3C illustrate a wavelet decomposition of an example signal andexample wavelet basis functions suitable for use with an embodiment ofthe invention;

FIGS. 4A-4B illustrate methods of creating a bandlimited noise octaveaccording to an embodiments of the invention;

FIG. 5 illustrates an example application of a method of creating abandlimited noise octave according to an embodiment of the invention;

FIG. 6A illustrates the frequency spectrum of an example bandlimitednoise octave according to an embodiment of the invention;

FIG. 6B illustrates the creation of an example noise function from twoor more noise octaves according to an embodiment of the invention;

FIG. 7 illustrates an optimized method for creating bandlimited noiseoctaves according to an embodiment of the invention;

FIG. 8 illustrates an example tiling of a pair of out-of-phase noiseoctaves suitable for use with an embodiment of the invention;

FIG. 9 illustrates a set of distributions for a noise octave, a noiseset, and a corrected noise set according to an embodiment of theinvention;

FIG. 10 illustrates an example application of relative noise octaveweights used to control the characteristics of a noise set according toan embodiment of the invention;

FIG. 11 illustrates a method for combining attribute data sets accordingto an embodiment of the invention;

FIG. 12 illustrates a frequency spectrum of an example combination ofattribute sets according to an embodiment of the invention;

FIGS. 13A-13B illustrate arbitrary tiling of noise octaves and otherdata according to an embodiment of the invention;

FIG. 14 illustrates an example approximation of the filter kernel usedfor typically rendering noise octaves according to an embodiment of theinvention;

FIGS. 15A-15B illustrate an example slicing a noise data and theproblems associated with the same according to a prior technique; and

FIG. 16 illustrates an example application of creating a bandlimitedslice of data from a higher dimensional set of data according to anembodiment of the invention.

DETAILED DESCRIPTION OF THE INVENTION

FIG. 1 illustrates a block diagram of a computer system suitable forimplementing an embodiment of the invention. FIG. 1 illustrates anexample computer system 100 capable of implementing an embodiment of theinvention. Computer system 100 typically includes a monitor 110,computer 120, a keyboard 130, a user input device 140, and a networkinterface 150. User input device 140 includes a computer mouse, atrackball, a track pad, graphics tablet, touch screen, and/or otherwired or wireless input devices that allow a user to create or selectgraphics, objects, icons, and/or text appearing on the monitor 110.Embodiments of network interface 150 typically provides wired orwireless communication with an electronic communications network, suchas a local area network, a wide area network, for example the Internet,and/or virtual networks, for example a virtual private network (VPN).

Computer 120 typically includes components such as one or more generalpurpose processors 160, and memory storage devices, such as a randomaccess memory (RAM) 170, disk drives 180, and system bus 190interconnecting the above components. RAM 170 and disk drive 180 areexamples of tangible media for storage of data, audio/video files,computer programs, applet interpreters or compilers, virtual machines,embodiments of the herein described invention including geometric scenedata, object data files, shader descriptors, a rendering engine, outputimage files, texture maps, and displacement maps. Further embodiments ofcomputer 120 can include specialized audio and video subsystems forprocessing and outputting audio and graphics data. Other types oftangible media include floppy disks; removable hard disks; opticalstorage media such as DVD-ROM, CD-ROM, and bar codes; non-volatilememory devices such as flash memories; read-only-memories (ROMS);battery-backed volatile memories; and networked storage devices.

Perlin noise is often used as an input to procedural data sets. FIG. 2Aillustrates an octave of Perlin noise 200 created according to a priormethod. Typically, a base resolution or spatial frequency is chosen forthe octave of Perlin noise. It should be noted that the term frequencyis used loosely throughout this application to mean either the result ofa Fourier transform (or other frequency domain transform) or theresolution of a set of data. A regular grid of sample points such aspoints 205, 210, 215, 220, and 225 are created. The distance betweenadjacent sample points is based upon the value of the base spatialfrequency. The value of the noise is set to zero at each sample point.Additionally, a randomly selected gradient value is assigned to eachsample point, such as gradients 230, 235, and 240 assigned to samplepoints 205, 215, and 225. Using the value of the noise function and thevalue of the gradient at each sample point, noise values 245 areinterpolated for the noise octave 200. In some implementations, thenoise values 245 are defined by a polynomial spline fitted to the samplepoint values and gradients. Although a one-dimensional noise octave isshown in FIG. 2A, this approach can be extended to generate noiseoctaves having two, three, or more dimensions.

A complete Perlin noise set is generated by combining two or more noiseoctaves having different base frequencies. To avoid aliasing andblurring, each noise octave should ideally be bandlimited to thefrequency range between its base frequency value and twice its basefrequency value. However, with Perlin noise, this condition is notsatisfied. FIG. 2B illustrates the frequency spectrum 265 of the examplenoise octave 200 discussed above. The frequency scale 250 is marked withthe value of the base frequency 260 and twice the base frequency 255.However, as can be seen in FIG. 2B, a substantial portion 270 of thefrequency spectrum 265 is below the base frequency 260 and a smallerportion 275 of frequency spectrum 265 is above twice the base frequency255. Relatively small changes in the slope of the noise octave, such asin portions 248 and 249 in FIG. 2A, introduce substantial low frequencycomponents that contribute to portion 270 of the frequency spectrum 265,while relatively “steep” changes in the slope of the noise octave, suchas portion 247 in FIG. 2A, may introduce high frequency components thatcontribute to portion 275. The presence of high and low frequencycomponents outside of the desired frequency band creates unwantedaliasing and blurring, respectively. In particular, the weakbandlimiting of low frequency components in Perlin noise introducessubstantial difficulties with its use. Additionally, because the Perlinnoise function typically passes through zero at every sample point,regular patterns are introduced into the noise function.

An embodiment of the invention uses function spaces generated by scalingfunctions to generate bandlimited noise octaves. Scaling functions arebasis functions that admit multiresolution analysis. A variety ofdifferent types of multiresolution analysis scaling basis functionsknown to those of skill in the art can be used with the presentinvention, including piecewise constant scaling functions, such as Haarbasis functions; piecewise polynomial scaling functions, such as uniformB-spline basis functions of any degree; bandlimited scaling functions,such as sinc functions; and Daubeschies scaling functions. Scaling basisfunctions such as Haar basis functions, uniform B-spline basisfunctions, and Daubeschies scaling functions are locally supported,i.e., these scaling basis functions have finite ranges. Other scalingbasis functions, such as the sinc function, have an infinite support,i.e. have an infinite range. As discussed in detail below, locallysupported scaling basis functions allows noise data to be reconstructedusing a small number of noise coefficients in the vicinity of the pointof evaluation.

In general, multiresolution analysis techniques such as wavelettransformations and Laplacian pyramids can be used to determine one ormore lower resolution versions of a set of data. With wavelettransformations, a higher resolution version of a data set can beconstructed from a lower resolution version of the data set and a set ofdetail coefficients. The set of detail coefficients are multiplied by aset of wavelet or detail basis functions to augment the vector space ofthe lower resolution version of the data set with the additionalresolution of the higher resolution version of the data set. Additionaldetail coefficients can be used to further reconstruct the set of dataat additional higher levels of resolution.

In general, a wavelet transformation of a set of data can be determinedby projecting the set of data onto a set of scaling basis functions anda set of wavelet basis functions. In an embodiment, this projection isperformed by taking the dot or scalar product of the set of data and thedual of each of the scaling and wavelet basis functions. The resultingprojection gives the value of one or more scaling coefficients (C) andone or more detail coefficients (D). For orthogonal wavelet basisfunctions, such as Haar basis functions (discussed below) or Daubeschiesbasis functions, the dual of a basis function is the same as the basisfunction itself. However, for non-orthogonal basis functions, such asB-spline basis functions, the scaling functions and their duals aredistinct.

There are a variety of different types of scaling and detail basisfunctions that can be used for wavelet transformations. One of thesimplest types is the Haar basis. The Haar basis is a set of stepfunctions and square waves. The Haar basis has the property that the setof scaling and wavelet basis functions for a given level are orthogonalto each other and to the scaling and wavelet basis functions for allhigher levels. FIG. 3A, discussed in detail below, illustrates the Haarscaling basis function for the N=0 level and the set of wavelet basisfunctions for the N=1, 2, and 3 levels of an example eight element dataset. By determining the scaling and detail coefficients associated withthese basis functions for a set of data, the set of data can bereconstructed at any resolution up to the maximum resolution of the setof data.

TABLE 1 Example Wavelet Transformation Maximum Level (N) C D Resolution3 1 3 3 9 8 10 5 17 — R 2 2 6 9 11 1 −3 1 6 R/2 1 4 10 2 1 R/4 0 7 3 R/8

An additional technique for determining a wavelet transformation of adata set using the Haar basis is to recursively average and differenceadjacent values of a data set. Table 1 illustrates a wavelettransformation of an example data set having the values [1,3,3,9,8,10,5,17]. This set of eight values (2³) corresponds to a level, N, of 3. Thedata set at level N=3 can be referred to as C₃ To determine the wavelettransformation of this set of data for level N=2, each of the adjacentdata values are averaged. This forms a set of scaling coefficients atthe level N=2, referred to as C₂. For example, the average of the firsttwo terms of C₃, 1 and 3, is 2, which is the first term of C₂.Similarly, the average of the 3 and 9 forms the second term, 6, of C₂.

Similarly, the difference between a pair of adjacent data values and theaverage its corresponding scaling function coefficient at the nextlowest level is determined to define a set of detail coefficients. Forexample, the first term of the set of detail coefficients at the N=2level, referred to as D₂, is 1, which is the difference between thesecond term of C₃, 3, and the corresponding first term of C₂, 2. Thisterm of D₂ is also the difference between the sum of the correspondingfirst term of C₂, 2, and the first term of C₃, 1. Similarly, thedifference between the third and fourth terms of the C₃ set and thecorresponding second term of the C₂ set is −3. In general, for a pair ofscaling coefficients at level N, a corresponding detail coefficient atlevel N−1 is the difference between a corresponding scaling coefficientat level N−1 and a left-side scaling coefficient of the pair at level N.Similarly, for a pair of scaling coefficients at level N, acorresponding detail coefficient at level N−1 is the difference betweenthe right-side scaling coefficient of the pair at level N and acorresponding scaling coefficient at level N−1.

This process of averaging and differencing can be repeated to determinescaling coefficients for successively lower resolution levels. The dataset can be reconstructed at any resolution level up to its originalresolution using the set of scaling coefficients and the set of detailcoefficients. At each level, the set of scaling coefficients representsa lower resolution version of the original data set. FIG. 3A illustratesa wavelet decomposition 300 of an example signal suitable for use withan embodiment of the invention. The wavelet decomposition 300 includes ascaling basis function 305 for resolution N=0, and wavelet basisfunctions D₀, 310; D₁, 315 and 320; and D₂, 325, 330, 335, and 340.Scaling basis functions for C₁, C₂, and C₃ have been omitted forclarity, but are similar to basis function 305. Multiplying each basisfunction by its associated coefficient and summing the results can beused to reconstruct a version of the data set at any resolution level.For example, the scaling coefficient of C₀ and detail coefficient setsD₀ can be used to reconstruct C₁, which is a version of the data set atresolution level N=1. Detail coefficient set D₁, in conjunction with C₀and D₀ or with C₁, can be used to reconstruct C₂, which is a version ofthe data set at resolution level N=2. Similarly, detail coefficient setD₂, in conjunction with either C₀, D₀, and D₁; C₁ and D₁; or C₂, can beused to reconstruct C₃, which is the original data set.

For any type of scaling basis functions, the projection of the originaldata set onto the scaling basis functions of a level result in the bestpossible representation of the data set at this resolution level. Forexample, the projection of a data set onto a set of spline scaling basisfunctions of a given resolution level results in a set of scalingcoefficients representing a least squares best fit representation of thedata set at this resolution level.

At each level, the set of scaling coefficients represents a lowerresolution version of the original data set. Each resolution level ofthe data set is bandlimited. For example, if the resolution of the dataset C₃ at resolution level N=3 is R, the resolution of this data set atresolution level N=2 is R/2. Similarly, the resolution of the data setat resolution level N=1 is R/4. The set of detail coefficients at eachlevel represent additional information that cannot be represented by thescaling coefficients at that level. In the case of orthogonal basisfunctions such as the Haar basis and other types of basis functions suchas a B-spline basis function, the detail coefficients of a givenresolution level represent all of the information in the next higherresolution level that cannot be represented at the given resolutionlevel.

If a basis function, such as a Haar or spline basis function, centeredat x=0 is represented as φ(x), then other vectors F(x) can berepresented as the linear combination of scaled versions of the basisfunction translated by integer amounts i. This can be expressed as

${F(x)} = {\sum\limits_{i}{f_{i}{{\phi\left( {x - i} \right)}.}}}$

The set of all vectors that can be expressed by varying the values ofthe coefficients f_(i) forms a vector space at a first resolution level,referred to as vector space S₀. As described above with the Haar basis,a set of basis functions at a second resolution level can be used torepresent additional detail of the function in a higher resolutionvector space, referred to as vector space S₁.

For some types of basis functions, the set of basis functions in S₁ canbe derived from the basis functions S₀ by reducing the width of thebasis function in S₀ in half and translating by half integer values.Thus, if the basis function in S₀ is φ(x), the basis function in S₁ isφ(2x−i). This relationship can be extended to any arbitrary resolutionlevel N, where the basis functions in vector space S_(N) are scaled inwidth by half and translated by half integer values from the basisfunctions of vector space S_(N−1).

All vectors in S₀ are also in S₁ if there are coefficient values p_(k)such that:

${\phi(x)} = {\sum\limits_{k}{p_{k}{{\phi\left( {{2x} - k} \right)}.}}}$If these coefficient values exist, then all vectors in S₀ are also in S₁and the basis function φ(x) is said to be refineable. However, not allvectors in vector space S₁ are in vector space S₀. Examples ofrefineable basis functions include piecewise constant scaling functions,such as Haar basis functions; piecewise polynomial scaling functions,such as uniform B-spline basis functions of any degree; bandlimitedscaling functions, such as sinc functions; and Daubeschies scalingfunctions.

FIG. 3B illustrates an example uniform quadratic B-spline basis function350. FIG. 3C illustrates the refineability of example spline basisfunction 350. Outline 375 corresponds with example spline basis function350 in vector space S_(N), corresponding with a resolution level of N.Example spline basis function 350 can be represented as the linearcombination of basis functions in vector space S_(N+1) that have beenreduced in width by half, offset along half-integers, and scaled inheight by appropriate values of p_(K).

In this example, the spline basis function 350 can be represented as thelinear combination of basis functions 380, 385, 390, and 395. Each ofthese basis functions 380, 385, 390, and 395 has a width that is half ofthat of basis function 350. Additionally, each of the basis functions380, 385, 390, and 395 is offset along a half integer from basisfunction 350. For example, basis function 380 is centered a half integerto the right of the center of basis function 350. Basis function 390 iscentered at 3/2 integers to the right of the center of basis function350. Similarly, basis function 385 is centered a half integer to theleft of the center of basis function 350 and basis function 395 iscentered at 3/2 integers to the left of the center of basis function350.

The basis functions 380, 385, 390, and 395 are scaled in height relativeto the height of basis function 350 according to their respective valuesof p_(k). For quadratic B-splines, the values of p_(k) are ¼, ¾, ¾ , and¼ for basis functions 395, 385, 380, and 390, respectively. The valuesof p_(k) for basis functions in S_(N+1) centered at all other locationsare zero, and thus these basis functions have been omitted from FIG. 3C.Points 396, 386, 381, and 391 illustrate the values of p_(k) for basisfunctions 395, 385, 380, and 390, respectively.

An embodiment of the invention uses these properties of the waveletbasis functions and their associated detail coefficients to createbandlimited noise octaves. FIG. 4 illustrates a method 400 of creating abandlimited noise octave according to an embodiment of the invention. Atstep 405, an unfiltered noise set, C_(N), is created. The unfilterednoise set can be created using any technique for generating random orpseudorandom data. The unfiltered noise set can be arranged as a one,two, three, or more dimensional set and can include any number ofelements. In a further embodiment, the unfiltered noise set is aninfinite set, portions of which are evaluated as needed by method 400.In an embodiment, the maximum spatial frequency of the unfiltered noiseset is specified by the Nyquist limit of the unfiltered noise set. Inanother embodiment, the maximum spatial frequency of the unfilterednoise set can be less than its Nyquist limit. In an embodiment, theunfiltered noise set will have a substantially flat frequency spectrum,equivalent to that of white noise. FIG. 5 illustrates an exampleapplication 500 of a method of creating a bandlimited noise octaveaccording to an embodiment of the invention. FIG. 5 shows an exampleunfiltered noise set 505 resulting from step 405.

Step 410 applies a filtering operation on the unfiltered noise set C_(N)to produce a lower resolution level of the unfiltered noise set C_(N−1)and a set of detail coefficients D_(N−1) representing the portion of theunfiltered noise set that cannot be represented by C_(N−1). In anembodiment, step 410 projects the unfiltered noise set C_(N) onto a setof scaling basis functions and wavelet basis functions for the N−1 levelof resolution. Step 410 can use any type of basis functions, including aHaar basis or a spline basis. In a further embodiment, step 410 can skipthe determination of C_(N−1) and only determine the set of detailcoefficients D_(N−1).

Returning to the example of FIG. 5, step 410 creates a set of detailcoefficients D_(N−1) 510 and/or a lower resolution level of theunfiltered noise set C_(N−1) 515. In this example, the unfiltered noiseset C_(N) is a two-dimensional set. As a result, the set of detailcoefficients 510 can be subdivided into a set of horizontal detailcoefficients, DX_(N−1), a set of vertical detail coefficients, DY_(N−1),and a set of diagonal detail coefficients DXY_(N−1).

Step 415 performs an inverse transformation on the set of detailcoefficients D_(N−1) alone, without the contribution of C_(N−1), todetermine the noise octave Noise_(N). In an embodiment, this isperformed by assuming the value of C_(N−1) is 0 for all of its terms.For example in FIG. 5, the value of C_(N−1) 520 is assumed to be 0. Inan example inverse transformation, using a Haar basis, the left andright terms of each pair of terms in the noise octave Noise_(N) is equalto the corresponding term of the set of detail coefficients D_(N−1)multiplied by −1 and 1, respectively. Continuing with this example, ifthe first term of the set of detail coefficients D_(N−1) is 1, then thefirst two terms of the noise octave Noise_(N) are −1 and 1,respectively. Similarly, if the second term of the set of detailcoefficients D_(N−1) is −3, then the third and fourth terms of the noiseoctave Noise_(N) are 3 and −3, respectively. Table 2 illustrates thevalues of an example noise octave Noise₃ created from the unfilterednoise set C_(N) shown in Table 1. Noise_(N) 530 in FIG. 5 is an examplenoise octave determined from the inverse transformation of detailcoefficient set 510 with the value of all elements of C_(N−1) assumed tobe 0.

TABLE 2 Example Wavelet Noise Octave Noise Octave Level (N) C D NoiseFrequency Band 3 1 3 3 9 8 10 5 17 — −1 1 −3 3 −1 1 −6 6 F to F/2 2 2 69 11 1 −3 1 6

As discussed above, because the set of detail coefficients D_(N−1)represent the portion of data set, in this case the unfiltered noise setC_(N), having frequency components between the maximum frequencies oftwo adjacent levels of resolution, the inverse transformation of thisset of detail coefficients D_(N−1) is a set of noise bandlimited betweenthese two frequencies.

There are numerous equivalent methods of determining a bandlimited noiseoctave. In an embodiment, the determination of the set of detailcoefficients D_(N−1) can be bypassed. FIG. 4B illustrates another method450 of determining a bandlimited noise octave.

Step 455 of method 450 creates an unfiltered noise set, C_(N), similarto that discussed above. Step 460 transforms the unfiltered noise setC_(N), into a lower resolution noise set C_(N−1). In an embodiment, step460 downsamples unfiltered noise set C_(N) by a factor of two to createC_(N−1). In an embodiment, step 460 averages two noise samples for eachdimension of noise set C_(N) to determine each noise sample in C_(N−1).In a more general embodiment, if the noise set C_(N) is expressed as thelinear combination of basis functions at resolution level N, step 460projects the samples of noise set C_(N) onto the basis functions ofresolution level N−1 to produce noise set C_(N−1). As noise set C_(N−1)has a lower resolution than noise set C_(N), some information from noiseset C_(N) is lost in the transformation to noise set C_(N−1). In afurther embodiment, the unfiltered noise set C_(N) can be filtered usinga spatially invariant or spatially variant filter to create C_(N−1). Inan embodiment, the maximum spatial frequency of the noise set C_(N−1) isspecified by the Nyquist limit of noise set C_(N−1). In anotherembodiment, the maximum spatial frequency of noise set C_(N−1) can beless than its Nyquist limit.

Step 465 upsamples or inverse transforms noise set C_(N−1) to resolutionlevel N. In an embodiment, step 465 scales noise set C_(N−1) by a factorof two in each dimension to create an upsampled version of noise setC′_(N−1). This is mathematically equivalent to assuming the detailcoefficients D_(N−1) are equal to zero. In another embodiment, step 465applies the inverse of the transformation or projection of step 460 tothe noise set C_(N−1) to create an upsampled version of the noise set,C′_(N−1).

Step 470 subtracts the unsampled noise set C′_(N−1) from the originalunfiltered noise set C_(N) that was created in step 455. The differencebetween these two noise sets is a bandlimited set of noise, which has afrequency range between the maximum frequency of level N−1 and level N.As discussed above, some information from C_(N) is lost in thetransformation from C_(N) to C_(N−1). This lost information correspondsto information in the frequency range greater than the maximum frequencyof resolution level N−1. By subtracting the upsampled noise set C′_(N−1)from noise set C_(N), this lost information in the frequency rangebetween the maximum frequency of resolution level N−1 and resolutionlevel N is isolated.

Table 3 illustrates the values of an example noise octave Noise₃ createdfrom the unfiltered noise set C_(N) shown in Table 1 using method 450.In this example, the unfiltered noise of resolution level 3 isdownsampled to resolution level 2, then upsampled to back to resolutionlevel 3. The upsampled version of the noise, C′₃, is subtracted from theunfiltered noise C_(N) to define a bandlimited noise octave.

TABLE 3 Example Wavelet Noise Octave Level (N) C C′ Noise = C − C′ 3 1 33 9 8 10 5 17 2 2 6 9 11 3 2 2 6 6 9 9 11 11 −1 1 −3 3 −1 1 −6 6

Methods 400 and 450 of creating a bandlimited noise octave aremathematically equivalent. In method 400, C_(N−1) is set to zero andD_(N−1) is inverse transformed to determine D_(N). In method 450,D_(N−1) is set to zero and C_(N−1) is inverse transformed and subtractedfrom C_(N). The methods 400 and 450 determine a set of noisecoefficients n_(i) describing a bandlimited noise octave. The value ofthe bandlimited noise octave at any location x (e.g. Noise(x)) isdefined as the linear combination of the noise octave's scaling basisfunctions weighted by the nearby noise coefficients n_(i). This may beexpressed as

${N(x)} = {\sum\limits_{i}{n_{i}{{\phi\left( {{2x} - i} \right)}.}}}$

Because of the properties of the basis functions used to constructbandlimited noise octaves, it can be shown that any bandlimited noiseoctave at resolution level N is orthogonal to the basis functions at allresolution levels less than N, such as resolution levels N−1, N−2, andso forth. This is true regardless of whether the scaling basis functionsat different resolution levels are orthogonal to each other, such asHaar scaling basis functions, or not orthogonal to each other, such asB-spline scaling basis functions.

Mathematically, the orthogonality property of bandlimited noise octavescan be expressed for a one-dimensional set of noise as∫Noise_(N)(2^(j)x−l)φ(x−i)dx=0 for all integers l and i when integer jis greater than or equal to zero. Integer j represents the resolutionlevel under evaluation. In this expression, j=−1 indicates the currentresolution level and increased values of j indicate lower resolutionlevels. Thus, if bandlimited noise octaves are constructed using waveletbasis functions for resolution level N, they can be scaled to any otherresolution level N′ and be guaranteed to have no effect on images at anyresolution level less than N′.

Expressed another way, if refineable scaling basis functions are used tocreate an unfiltered noise set in vector space S_(N) and a filteredversion of the noise set in lower resolution vector space S_(N−1), thenvector space S_(N−1) is contained within vector space S_(N), due to therefineability property of the scaling basis functions. Morever, thebandlimited noise set is found in a complement of vector spaces S_(N−1)and S_(N), such that at least a portion of the bandlimited noise setcannot be represented in vector space S_(N−1). In a further embodiment,the bandlimited noise set is found in an orthogonal complement of vectorspaces S_(N−1) and S_(N), such that no portion of the bandlimited noiseset can be represented in vector space S_(N−1).

FIG. 6A illustrates the frequency spectrum 600 of an example bandlimitednoise octave according to an embodiment of the invention. Frequencyspectrum 600 includes the spectrum 605 of the unfiltered noise setC_(N). Spectrum 605 has a maximum frequency of F, 610. After filteringthe noise set as described in step 410, the C_(N−1) set has a spectrum615. Spectrum 615 has a maximum frequency F/2, 620. The set of detailcoefficients D_(N−1) represent the information in Cn that is in spectrum625. Spectrum 625 is bounded by frequencies F/2, 620, and F, 610. As aresult, the inverse transformation of the set of detail coefficientsD_(N−1) will also be substantially bandlimited between thesefrequencies, with minimal spectral components outside of this frequencyband due to numerical limits of calculation and other minor errors andapproximations.

In an embodiment, the frequency spectrum of the bandlimited noise isapproximately zero outside of the range between frequencies F/2, 620 andF, 610. In another embodiment, the frequency spectrum of the bandlimitednoise is approximately flat between frequencies F/2, 620 and F, 610 anddiminishes abruptly outside of this range. In a further embodiment, thebandlimited noise diminishes from a significant magnitude toapproximately zero within a power of two range from frequencies F/2, 620and F, 610. In still another embodiment, the frequency spectrum of thebandlimited noise is such that the sum of the frequency spectrum of thefirst bandlimited data set and a copy of the frequency spectrum of thefirst bandlimited data is substantially flat, if the copy of thefrequency spectrum is translated in frequency up or down by thedifference between frequencies F, 610 and F/2, 620.

A complete noise function can be created by combining two or morebandlimited noise octaves created as described in method 400. Noiseoctaves can be generated at different frequency bands or at the same oroverlapping frequency bands. Noise octaves can be created from the sameunfiltered noise function or different unfiltered noise functions. Thenoise set can be used to define attribute data of objects, eitherdirectly or as an input to an algorithm or procedure.

FIG. 6B illustrates the creation of an example noise function 650 fromtwo or more noise octaves according to an embodiment of the invention.Noise function 650 is created from the weighted sum of noise octaves655, 660, 670, and 675. The weights associated with each noise octave,such as 4 with noise octave 655, can be chosen to emphasize ordeemphasize portions of the spectrum of noise function for aestheticpurposes. Additionally, noise octaves can be removed from the noise setto bandlimit the noise or any attribute data created from the noise, sothat aliasing and blurring do not occur. An additional embodimentattenuates the highest and/or lowest frequency noise octaves in a noisefunction based upon a center frequency of the noise function, forexample using a cubic function. This smoothly fades noise bands into andout of the noise function, preventing abrupt transitions and poppingartifacts when the noise function is applied to surfaces where thecenter frequency gradually changes, such as a ground plane extending tothe horizon.

Large objects and complex scenes, for example those shown in landscapes,often use a large or even infinite set of noise data to determine itsattribute data. In these cases, portions of the infinite unfilterednoise set can be generated procedurally on demand as the correspondingportions of the object are being rendered, for example using a random orpseudorandom number generator. To create noise octaves on demand for agiven point on an object, the values of the unfiltered noise set must bedetermined for a number of points surrounding the given point of theobject. Typically, the number of surrounding points must be sufficientlylarge so that the excluded portions of the unfiltered noise set onlycontribute negligibly, if at all, to the value of the noise octave atthe given point.

For example, to determine the value of a one-dimensional noise octavefor a given point on demand, the unfiltered noise values ofapproximately 40 surrounding points must be processed. Thus, in anembodiment applying method 400, the determination of the noise value ondemand for a single given point requires generating unfiltered noisevalues for 40 points, filtering the set of 40 unfiltered noise values todetermine a set of 20 detail coefficients, and then inverse transformingat least one of the set of detail coefficients to determining the valueof the noise octave at the given point. This number of points isselected because it encompasses the unfiltered noise values inapproximately 99% of the area of an example quadratic B-spline basisfunction. For other types of basis functions and/or different accuracyrequirements, a different number of unfiltered noise values can beevaluated to determine the value of a point in a noise octave.

In some applications, noise octaves can be precalculated and cached.However, sample points on an object during rendering are sometimes sofar apart that they fall outside of the range of cached noise octavevalues, especially when rendering distant objects. Furthermore, thecomputational requirements increase exponentially for noise sets havingmore than one dimension. Continuing with the above example, if aone-dimensional noise octave requires the evaluation of 40 surroundingpoints of the unfiltered noise data, a two-dimensional noise octaverequires the evaluation of 1600 surrounding points and athree-dimensional noise octave requires the evaluation of 64000surrounding points.

To reduce the computational requirements in computing the value of noiseoctaves on demand, a further embodiment precalculates the values ofmultiple sets of noise octaves, referred to as noise tiles. Noise tilescan be reused to determine the values of the noise functions at manydifferent points. For example, a tile of random numbers, such as thatdefined in step 405 or 455, can be applied to any portion of an infiniteset, representing a line, plane, volume, or higher-dimensional set.Bandlimited noise data for this portion of an infinite set can bedefined using method 400, 450, or equivalents thereof. In determining abandlimited noise values from a tile of random numbers, an embodiment ofthe invention can assume that the tile of random numbers repeats whenthe downsampling or upsampling filter crosses tile boundaries. Theresulting bandlimited noise set or noise tile can be tiled over theinfinite set without the appearance of seams. However, a repeatingpattern can be discernable from noise tiles, particularly if the noisetile is relatively small compared with the field of view. Embodimentsdiscussed below can reduce or eliminate the appearance of patterns whentiling or otherwise combining sets of bandlimited noise.

FIG. 7 illustrates an optimized method 700 for creating bandlimitednoise octaves according to an embodiment of the invention. Step 705generates at least two libraries, M and N, of bandlimited noise octavesusing the method 400 discussed above. Each library includes one or morebandlimited noise octaves. The noise octaves of each library can eachhave any number of dimensions and can be based on unfiltered noise setsof any arbitrary size. In an embodiment, the unfiltered noise sets usedfor the M and N libraries are differently sized to create substantialphase differences when tiling noise octaves from the M and N libraries.This can be selecting sizes for the unfiltered noise sets that maximizetheir least common multiple. Additionally, unfiltered noise sets havingsizes that are even numbers is convenient for determining bandlimitednoise octaves. For example, noise octaves in the M library can bederived from an unfiltered noise set of with 42 elements, while thenoise octaves in the N library can derived from an unfiltered noise setwith 38 elements.

FIG. 8 illustrates an example tiling 800 of a pair of out-of-phase noiseoctaves suitable for use with an embodiment of the invention. Tiling 800includes multiple copies of noise octave M, 805 a, 805 b, and 805 c,arranged end to end. Tiling 800 includes multiple copies of noise octaveN, 810 a, 810 b, and 810 c, similarly arranged. In this example, noiseoctaves 805 and 810 are one dimensional sets of noise. The set 815 ofmultiple copies of noise octave M forms a line, which can be extended asneeded for any arbitrary length. Similarly, the multiple copies of noiseoctave N form line 820. If the noise octaves are two dimensional sets ofnoise, then each set of multiple copies of a noise octave forms a plane.In embodiments, tiling 800 can be extended to noise octaves of three ormore dimensions.

In the example tiling 800, noise octaves M and N are differently sizedto create phase differences when tiling. For example, although the firstelements of octaves 805 a and 810 a will be aligned, the beginning ofoctave 805 b will be aligned with the end of octave 810 a, at point 825a. Similarly, the beginning of octave 805 c will be aligned with the endof octave 810 b, at point 830 b. In this example, point 830 b is to theleft of point 825 b, which corresponds to point 825 a of octave 810 a.

Step 710 selects a point at which the noise value is to be determined.This point can be determined by a renderer or any other hardware and/orsoftware system. Step 710 selects at least one noise octave from each ofthe libraries. In an embodiment, the selected noise octaves have similaror the same frequency bands. In an embodiment, each library includes twoor more noise octaves for each frequency band and step 710 randomly orpseudorandomly selects one or more noise octaves from the library.

Step 720 determines indices or coordinates corresponding to the selectedpoint in each of the selected noise octaves. In an embodiment, it isassumed that each of the selected noise octaves is part of an infinitelytiled set, for example as described in FIG. 8. In this embodiment, thecoordinates of the selected point are determined within this infinitelytiled set. For example, the coordinates of the selected point withinthis infinitely tiled set can be the coordinates of the point in a localcoordinate system, such as a texture coordinate system. In anembodiment, the coordinates of the point in each of the selected noiseoctaves is determined by the modulo division of the point's coordinatesin the infinitely tiled set by the size of each dimension of each noiseoctave.

Step 725 retrieves the values of the selected noise octaves at thelocations specified by the sets of coordinates of the selected point inthe selected noise octaves. The values of the selected noise octaves arethen added or otherwise combined to create a combined noise octave valuefor the selected point. Because the selected noise octaves havedifferent sizes, thereby tiling out of phase, the appearance of regularpatterns due to repeating noise octave values is reduced. In anembodiment, the combined noise octave value can be normalized or scaledto fall within a desired numerical range.

Step 730 repeats steps 715 through 725 to determine one or moreadditional combined octave values at other frequency bands. In anembodiment, the number of other frequency bands considered is limited bythe maximum spatial bandwidth supported by the renderer. Step 730determines the value of the noise function at the selected point bycombining the combined noise octave values from two or more frequencybands. As discussed in embodiments detailed below, the noise functioncan be specified as a weighted combination of combined noise octavevalues, similar to that discussed with reference to FIG. 6B. In anembodiment, the coordinates of the selected point are scaled to accountfor the different resolution of each noise octave.

One problem that can occur when combining random or noise data ischanges to the distribution of the values of the combined result. Forexample, if two functions have random output distributions, the outputdistribution of the combination of these functions will be theconvolution of their respective output distributions. As a result, thedistribution of the combined functions will be biased towards someoutput values.

FIG. 9 illustrates a set of distributions for a noise octave, a noiseset, and a corrected noise set according to an embodiment of theinvention. Distribution 900 is an example output distribution of arandom function. In this example, the distribution is flat, indicatingthat function has a equal probability of outputting any value between −1and 1. If two functions having distributions such as distribution 900are averaged, the resulting function has a distribution 905. As can beseen in distribution 905, the function has a high probability ofoutputting values around 0 and a very low probability of outputtingvalues near −1 or 1.

For example, consider if the first distribution 900 has only tendiscrete values (0.05, 0.15, 0.25, 0.35, 0.45, 0.55, 0.65, 0.75, 0.85,and 0.95). This distribution has a 10% probability of having a valuebetween 0 and 0.1. If two functions with distribution 900 are averaged,then according to the resulting distribution 905, the probability ofhaving a value between 0 and 0.1 substantially reduced. There is only a1% chance that both functions have a value of 0.5 (10% times 10%), andonly a 2% chance that one function has a value of 0.5 and the otherfunction has a value of 0.15. The combined probability of the functionwith distribution 905 having a value between 0 and 0.1 is thus 3%, whichis substantially less than the 10% probability in distribution 900.

An embodiment of the invention performs a distribution correction phaseafter combining two or more noise octaves as described above. In anembodiment, a distribution is associated with each noise octave. Thedistribution can be determined by analyzing each noise octave. As aresult, the distribution correction phase can determine the distributionof the combined noise function.

In the embodiments described above, the creation of bandlimited noiseoctaves from unfiltered set of noise data involves sequences of linearoperations on the data. If the unfiltered set of noise data follows aGaussian distribution of one or more random variables, then thebandlimited noise octaves will also be a Gaussian random variable.

One technique for determining the distribution of the combined noisefunction begins by representing noise values from bandlimited noiseoctaves as linear combinations of the unfiltered noise data. Given a setof K unfiltered noise values x, a bandlimited noise octave N can berepresented as:

$N = {\sum\limits_{i = 1}^{K}{w_{i}{x_{i}.}}}$This bandlimited noise octave N has a variance σ_(N) ², which can beexpressed in terms of the variance σ_(x) ² of the set of K unfilterednoise values x as

$\sigma_{N}^{2} = {\sigma_{x}^{2}{\sum\limits_{i = 1}^{K}{w_{i}^{2}.}}}$The set of weights w_(i) are derived from the operations used to createthe bandlimited noise octave from unfiltered noise data, such as theforward and reverse transformations of method 400 or the downsampling,upsampling, and subtracting operations of method 450. In an embodiment,the variance σ_(N) ² can be determined analytically using the set ofweights w_(i) and the variance σ_(x) ² of the unfiltered noise set. Inanother embodiment, the variance σ_(N) ² can be determined numericallyby analyzing the noise octave using typical statistical analysistechniques known in the art.

As described above, when two or more noise octaves are combined to forma set of bandlimited noise, the variance of the combined bandlimitednoise set changes further. Once the distribution of the noise functionis determined, a distribution correction can be applied to the noisefunction so that it has a desired distribution. The desired distributioncan be a flat, white noise distribution, such as distribution 910, aGaussian distribution, or any other arbitrary distribution. If theinitial noise function has a Gaussian distribution, a new Gaussiandistribution can be specified by scaling the variance of the initialnoise function to the variance of the new Gaussian distribution. For awhite distribution, an erf function can be used. Alternatively, a lookuptable or other type of mapping can be used to create a noise functionwith any arbitrary distribution from an initial noise function.

In an embodiment, a user can specify the desired distribution, forexample by inputting any arbitrary distribution function or byspecifying desired statistical properties of the noise function, such asthe mean and standard deviation. Thus, users can create a noise functionthat outputs ranges of values according to user-specified probabilities,thereby enabling precise control of the characteristics of input datafor procedural data sets. In an embodiment, the distribution correctionphase creates a mapping of values from the uncorrected noise function tovalues of the corrected noise function. The distribution correctionphase applies this mapping to the noise function to create a correctednoise function with the desired distribution.

As discussed above, noise functions can be created from weightedcombinations of different noise octaves. Typically, these weights areabsolute and are assigned to a fixed spatial frequency range. In someapplications, it is desirable to view an object or surface at a largerange of scales. For example, an animation may begin with a view of anentire planet and then zoom in to show a single blade of grass. Becausethe weights associated with the noise functions and other data have anabsolute weighting, if the zoom is centered on a relatively dark area ofthe noise function, the image will remain dark forever. As a result, thefine details are obscured.

In a further embodiment, the weights used to combine noise octaves candepend on the resolution scale or level of detail at which the noisefunction is viewed. These types of weights are referred to as relativeweights. As the level of detail changes, for example as the viewer movestowards a given point of the noise function, these weights are changeddynamically. This enables lower frequency noise octaves to be graduallyfaded out as the camera zooms in, which enhances the visibility of thefine or high frequency detail. This effect approximates the adaptationof human visual perception to the average intensity of an image. Inanother embodiment, absolute weights that are independent of theresolution scale can be used to combine noise octaves. In still afurther embodiment, both absolute and relative weights can be used tocombine noise octaves.

FIG. 10 illustrates an example application of relative noise octaveweights used to control the characteristics of a noise functionaccording to an embodiment of the invention. A frequency spectrum 1000includes noise octaves 1005, 1010, 1015, 1020. A relative weightingfunction 1025 is associated with the set of noise octaves. A centerfrequency 1030 is selected within the frequency spectrum 1000. Thecenter frequency 1030 can vary according to the desired resolution scaleor level of detail of the combined noise function. In an embodiment, thecenter frequency can be determined from a level of detail calculation,such as the size of a pixel or sub-pixel sample at a given viewingposition projected on to the combined noise function in world space.

The position of the relative weighting function 1025 within thefrequency spectrum 1000 is centered at the center frequency 1030. Todetermine the noise function at a given center frequency 1030, eachnoise octave is weighted according to the value of the relativeweighting function 1025. For example, noise octave 1005 is weightedaccording to the value of the relative weighting function at point 1035.Similarly, the weights of noise octaves 1010, 1015, and 1020 aredetermined by the values of the relative weighting function 1025 atpoints 1040, 1045, and 1050, respectively. In an embodiment, the valueof the relative weighting function 1025 is zero for all noise octavesabove the center frequency 1030.

The noise octaves 1005, 1010, 1015, and 1020 are then combined accordingto their assigned weights. As the center frequency 1030 changes positionwithin the frequency spectrum 1000, the relative weighting function 1025changes position to follow the center frequency 1030. As a result, theweights associated with each noise octave change as the relativeweighting function 1025 changes position.

In a further embodiment, the techniques described above for creatingbandlimited noise octaves can be extended to combine attribute data setsfrom a variety of different sources. The attribute data sets can definegeometry, for example expressed as displacement maps; and color,lighting, and shading attributes, for example expressed as texture,light, and shadow maps. The attribute data sets can be defined as theoutput of a procedure or defined explicitly, for example as an imagecreated by an artist or other user.

FIG. 11 illustrates a method 1100 for combining attribute data setsaccording to an embodiment of the invention. Two or more attribute datasets to be combined are sent to method 1100. Step 1105 transforms eachattribute data set into a set of bandlimited octaves, using techniquesdescribed above with reference to method 400 or method 700. In a furtherembodiment, either of these methods can be used depending upon the typeof attribute data set. For example, if the attribute data set isdescribed as a function including noise or other data as its input, thenmethod 700 can be applied in step 1105 to reduce processing time. Asmethod 700 determines the value of bandlimited octaves of noise or otherdata on demand, portions of method 700 will be used to determine thevalue of the attribute data set in step 1125, discussed below. Forattribute data sets defined explicitly, method 400 can be used todetermine bandlimited octaves for the entire attribute data set.

Step 1110 selects one or more points at which to evaluate thecombination of the attribute data sets. In an embodiment, step 1110selects points specified by a renderer that is rendering all or a partof an object associated with the attribute data sets.

At step 1115, an output frequency band is specified. In an embodiment,the output frequency band is specified by a renderer based upon size ofpixel or sub-pixel samples projected onto the attribute data sets at agiven point. When different points are rendered, or when the attributedata set changes position relative to the camera, the output frequencyband will change accordingly.

Step 1120 selects the bandlimited octaves associated with each attributedata set, if any, that fall within the output frequency band. In somecircumstances, an attribute data set may have no bandlimited octaveswithin the specified output frequency band. This may occur if anattribute data set specifies high frequency data that is above theoutput frequency band.

For each bandlimited octave selected in step 1120, step 1125 determinesthe coordinates corresponding to the selected point and retrieves thevalue of the attribute data set at that point.

Step 1130 combines the value from each of the selected bandlimitedoctaves into one or more combined values. In an embodiment, at leastsome of the attribute data sets associated with an object are combinedaccording to a set of weights. Step 1130 combines the data from theselected bandlimited octaves in accordance with a set of weightsassociated with their corresponding attribute data sets. Because thedata associated with each attribute data set may be represent differenttypes of attributes, not all of the data is combined directly. Forexample, attribute data defining the geometry of an object would not becombined with attribute data representing color. However, if two or moreattribute data sets include data of the same type, such as a proceduraltexture and a texture map both defining color data of an object, thenstep 1130 can combine data from selected noise octaves of theseattribute data sets according their associated weights.

Optional step 1135 determines if there are any additional points to beevaluated in the attribute data sets. If so, then method 1100 returns tostep 1110 to select the next point.

FIG. 12 illustrates a frequency spectrum 1200 of an example combinationof attribute data sets according to an embodiment of the invention. Inthis example, a texture map 1205, a displacement map 1210, a proceduraltexture 1215, and a noise function 1220 are associated with an object.Each of these attribute data sets is defined as one or more bandlimitedoctaves. For example, texture 1205 is represented by octaves 1225 and1230. Displacement map 1210 is represented by octaves 1235, 1240, and1245. Method 1100 determines an output frequency band for one or morepoints of an object and selects only the octaves of attribute datawithin this band. As a result, the attribute data sets associated withobjects are always sampled correctly, without any artifacts fromover-sampling or under-sampling.

As described above, precomputed noise tiles can be used to decreaseevaluation time. One problem with using precomputed noise tiles is theappearance of regular patterns at tile boundaries. The embodiments ofFIGS. 7 and 8 use out of phase combinations of noise tiles to hide thesepatterns. In another embodiment, noise tiles or other types of attributedata sets can be overlapped slightly and blended to hide tileboundaries. FIG. 13A illustrates an example 1300 of noise tilingaccording to an embodiment of the invention. In example 1300, threenoise tile 1305, 1310, and 1315 arranged so that there is a slightoverlap between the noise tiles. It should be noted that noise tiles1305, 1310, and 1315 may be arranged either in a regular grid pattern orin an arbitrary and/or irregular positions.

Noise tiles 1305 and 1310 include an overlap region 1312. Similarly,noise tiles 1305 and 1315 include overlap region 1317 and noise tiles1310 and 1315 include overlap region 1319. It should be noted thatoverlap regions 1312, 1317, and 1319 do not have to be the same size.

Blending operations are performed in each overlap region to hide tileboundaries. For example, the values of noise tiles 1305 and 1310 areblended in overlap region 1312. Embodiments of the invention can use anytype of linear or non-linear blending technique to combine data fromnoise tiles in the overlap regions. Typically, blending techniques scaleeach value of a tile based at least in part on its relative locationwithin the tile and decrease smoothly toward zero at the tile boundaryto hide the tile boundary.

Although overlapping and blending noise tiles and other attribute datasets can hide tile boundaries, the blending operation reduces thestatistical variance of the combined noise data, which in itself makesthe tile boundaries visible. This appears as reduced contrast in theoverlap regions along noise tile boundaries. For example, if at a givenlocation in the overlap region, the value from a first noise tile isscaled by α, the value from a second noise tile is scaled by 1−α, andthen scaled values are combined to create a blended noise value, thevariance of blended noise value at this location of the overlap regionis reduced by a factor of α²+(1−α)².

An embodiment of the invention compensates for this reduced variance byscaling noise values in overlap regions by the inverse of the reductionin variance. For example, if two tiles overlap and the mean of noisevalues in noise tile is zero and if the noise value in an overlap regionis scaled by α in one tile and by 1−α in the other tile, then the noisevalue can be divided by the square root of α²+(1−α)² to compensate forthe reduced variance. For non-zero mean noise values, the variance isadjusted by moving noise values in the overlap region closer to the meanby the square root of α²+(1−α)²

Noise tiles of two or more dimensions can be blended in a similarmanner. Normally, the weights of noise values from any number ofdifferent noise tiles overlapping at a location will sum to one.However, as discussed above, this results in a noise variance reductionthat is typically the sum of the squares of the noise tile values'weights used to determine the combined noise value at that location. Tocompensate for the reduction in variance, an embodiment divides eachnoise tile's weight by the square root of the noise variance reduction.After correcting for the variance reduction, the sum of the weights ofnoise values for different noise tiles will not equal one; however, thevariance of the combined noise value will be the same as the variance ofthe noise tiles.

Noise values can be scaled by α and variance compensation values at thetime when noise values from two or more noise tiles are combined. In afurther embodiment, if the pattern of tile overlaps are consistent sothat α values for noise values in a noise tile are constant, then thenoise values can be premultiplied by their appropriate variancecompensation values and α values at the time of the noise tile iscreated, eliminating the need to perform this calculation later whennoise tiles are combined.

FIG. 13B illustrates a method 1350 of combining noise tiles. Step 1355generates one or more bandlimited noise tiles as described above. Noisetiles can include bandlimited noise data in one, two, three, four, ormore dimensions. Step 1355 may create many different noise tiles thatcan be combined. In an embodiment, step 1355 decreases memoryrequirements by only generating a single noise tile. To provide avariety of different noise patterns, step 1355 would typically use a setof different noise tiles selected with a hash function. If memory spaceis a concern, another embodiment creates variations of the single noisetile by permuting the indices of the noise tile (for example, swappingx, y, and z indices into a three-dimensional noise tile) and by steppingthrough the noise tile in forward or reverse order. From a singlethree-dimensional noise tile, step 1355 can derive 48 differentvariations of the noise tile. In another embodiment, techniques such asWang tiles can tile space nonperiodically with fixed tile boundary pairsso the boundary matching can be precomputed.

Step 1360 selects two or more noise tiles (or variations of the samenoise tile) to be combined. Step 1360 also determines the size andlocations of overlap regions based on the positions of the noise tilesrelative to each other.

Step 1365 corrects for the reduced variance in the overlap region causedby blending. As described above, the reduction in variance for noisevalues can be determined by the value to be used in blending. Step 1365can then scale noise values to compensate for this reduction invariance. Embodiments of the invention can scale noise values tocompensate for reductions in variance at the time the noise tile iscreated or at the time the blending operation is performed. Step 1370then scales noise values according to their blending values and combinesnoise values from two or more noise tiles.

Typically, a renderer applies a filter kernel to turn a portion of ascene into a pixel value. If the scene at a position x is represented asa function S(x), the value of a pixel i can be expressed as

Pixel_(i) = ∫S(x)K(x − i)𝕕x,where K(x−i) is the renderer's filter kernel centered around pixel i. Inthe case of a bandlimited noise function, S(x) includes N(x), as definedabove. Typically, renderers approximate this integral using quadrature,such as the weighted sum of discrete values of the scene function. Inthe case where the scene function S(x) at position x is composed, atleast in part, by a bandlimited noise function N(x), this approximationcan be expressed as

${{Pixel}_{i} \approx {\sum\limits_{q = 1}^{Q}{B_{q}{N\left( x_{q} \right)}}}},$where Q is the number of samples evaluated by the renderer for eachpixel and the quadrature weights B_(q) corresponding to the renderer'sfilter kernel.

If the renderer evaluates a large number of samples of the bandlimitednoise function N(x) in the vicinity of a pixel (i.e. Q is relativelylarge), then the above expression

${Pixel}_{i} \approx {\sum\limits_{q = 1}^{Q}{B_{q}{N\left( x_{q} \right)}}}$is a good approximation of the average value of the bandlimited noisefunction N(x) in the vicinity of the pixel. However, if the renderer'ssampling rate is relatively low, then the expression

${Pixel}_{i} \approx {\sum\limits_{q = 1}^{Q}{B_{q}{N\left( x_{q} \right)}}}$does not give a good approximation of the average value of thebandlimited noise function N(x) in the vicinity of the pixel. Tocompensate for the renderer's limitations noise function N(x) can bechanged to Ñ(x_(q)). This can be expressed as

${{Pixel}_{i} \approx {\sum\limits_{q = 1}^{Q}{B_{q}{\overset{\sim}{N}\left( x_{q} \right)}}}},$where Ñ(x_(q)) is a weighted average of N(x) in the vicinity of x_(q).

In an embodiment, Ñ(x_(q)) is equal to the bandlimited noisecoefficients n_(i) times the convolution of the renderer's filter kernelwith the noise function's scaling basis function. For example, if thenoise function's scaling basis function and the renderer's filter kernelare both quadratic B-splines, then the result of this convolution willbe a quintic function. For other types of scaling basis functions and/orrenderer filter kernel's, different functions will result.

In yet a further embodiment, when both the noise function's scalingbasis function and the renderer's filter kernel are quadratic B-splines,the quintic function in the Ñ(x_(q)) can be approximated by a widenedversion of the renderer's filter kernel function B(x) or the noisefunction's scaling basis function. For example, a widened version of aquadratic B-spline basis function B(0.5(x−x_(q))) covers six noisecoefficients and is a good approximation of the integralPixel_(i)=∫S(x)K(x−i)dx at the highest resolution of the noise set, ifboth the noise function's scaling basis function and the renderer'sfilter kernel are quadratic B-splines. For lower resolution levels ofnoise octaves, narrower versions of the quadratic B-spline basisfunction covering fewer noise coefficients can be used.

FIG. 14A illustrates an example application of the evaluation of abandlimited noise function according to an embodiment of the invention.In this embodiment, point 1405 is the point of evaluation for thebandlimited noise function. A quadratic B-spline basis function 1410 iscentered around point 1405. Quadratic B-spline basis function 1410 canbe similar to the basis function used to express the bandlimited noisefunction or different weights that provides higher order accuracy can beused, such as a Gaussian quadrature weights.

Basis function 1410 covers three noise coefficients, 1417, 1419, and1421. The value of the noise function at point 1405 is represented asthe linear combination of noise coefficients 1417, 1419, and 1421, whereeach noise coefficient is scaled according to the value of the basisfunction 1410 at the location of the noise coefficient. For example, thevalue of the noise function at point 1405 is composed primarily of noisecoefficient 1417, because the basis function 1410 is near its peak valueat the location of noise coefficient 1417.

As discussed above, a widened version of a basis function can serve as agood approximation of the integral of a filter kernel applied to thenoise function. In the example of FIG. 14A, a widened basis function1415 centered at point 1405 is used to approximate the integral of thefilter kernel applied to the noise function at point 1405. In anembodiment, widened basis function 1415 is a wider version of basisfunction 1410. In alternate embodiments, widened basis function 1415 isa wider version of a different type of basis function. Widened basisfunction 1415 covers six noise coefficients. In addition to noisecoefficients 1417, 1419, and 1421, the widened basis function 1415 alsocovers noise coefficients 1423, 1425, and 1427. Thus, the value of theintegral centered at point 1405 can be approximated by the linearcombination of noise coefficients 1417, 1419, 1421, 1423, 1425, and1427, with each noise coefficient weighted by the value of the widenedbasis function 1415 at the location of the noise coefficient.

As discussed in detail below, when the noise coefficients are part of anoctave of bandlimited noise data, the contribution of other noisecoefficients located outside of the widened basis function 1415 to thevalue of the filter kernel integral at point 1405 is negligible or zero.Thus, the frequency spectrum of bandlimited noise data diminishes at thesame or similar rate as the filter kernel, such that the bandlimitednoise data is negligible or zero at the point of aliasing.

In an embodiment, the use of a widened basis function to approximate anintegral of noise coefficients around a point can be extended to two,three, four, or more dimensions. FIG. 14B illustrates an exampleapplication of the evaluation of a bandlimited noise function accordingto an embodiment of the invention. In this example, a two-dimensionalset of noise coefficients 1450 is illustrated. A point 1455 is selectedas the point of evaluation. The value of the noise function at point1455 can be determined by applying a basis function in two dimensions,such as a quadratic B-spline basis function, centered around point 1455to the set of noise coefficients 1450. Boundary 1460 represents theboundary of a two-dimensional basis function similar to basis function1410 discussed above, but extended to higher dimensions.

Similarly, the integral of noise coefficients around point 1455 can beapproximated by a wider version of a basis function. Boundary 1465represents the boundary of a two-dimensional widened basis functionsimilar to basis function 1415 discussed above, but extended to higherdimensions.

In one application of noise functions, noise functions are used as inputdata for creating procedural data sets. For example, a three-dimensionalset of noise data is used to create a procedural texture map for athree-dimensional volume, such as a cloud. In many applications, such asapplying a texture map to the surface of a three-dimensional object, theset of noise data is sliced or divided into a lower dimensional set ofnoise data.

For example, a two-dimensional plane or slice of noise data may beextracted from a three or more dimensional set of noise. FIG. 15Aillustrates an example noise set 1500 of three dimensions from which atwo-dimensional slice 1505 of noise data is extracted. Thetwo-dimensional slice 1505 can be used as input data for creatingprocedural data sets, such as procedurally generated texture maps. Insome applications, the two-dimensional slice 1505 is coincident with asurface of a three-dimensional model to which the procedural data setwill be applied.

As discussed above, bandlimited noise sets are useful in preventingaliasing or blurring artifacts. Unfortunately, it has been observed thata lower-dimensional slice of noise data extracted from a higherdimensional set of bandlimited noise data is not bandlimited. If examplenoise set 1500 is a bandlimited noise set, created as described above,two-dimensional slice 1505 will not be bandlimited.

FIG. 15B illustrates the reason why slices of bandlimited noise data arenot bandlimited as well. FIG. 15B illustrates the Fourier transform 1525of an example bandlimited noise set. For clarity, this example is for aone-dimensional slice through two-dimensional data. Fourier transform1525 illustrates the frequency components of the bandlimited noise setin the X direction 1527 and the Y direction 1529.

In FIG. 15B, the darker regions indicate greater magnitudes forfrequency components while the white regions indicate a zero ornegligible magnitude for frequency components. As can be seen from theFourier transform 1525, the center region 1530 has almost no spectralenergy. This is the result of the noise set being bandlimited so thatthe frequency components below a minimum frequency are negligible; inthis example those minimums are Fmin_(X) and Fmin_(Y).

FIG. 15B also illustrates the Fourier transform of a slice 1540 of thebandlimited noise set, analogous to as slice 1505 discussed above.According to the Fourier slice theorem, the value of any point of theFourier transform of the slice 1540 is equal to the integral of thenoise set on a line perpendicular to the slice and passing through thepoint.

For example, the value of the Fourier transform of the slice 1540 atpoint P 1545 is equal to the integral of the Fourier transform of thenoise set 1525 along line 1550. If slice 1540 were bandlimited, then thevalue of its Fourier transform at point P 1545, which lies in region1530, would be zero. However, according to the Fourier slice theorem,the value of the Fourier transform of slice 1540 at point P 1545 is theintegral of the Fourier transform of the bandlimited noise set 1525along line 1550. Line 1550 includes segment 1555, in which the value ofFourier transform 1525 is equal to zero. Unfortunately, line 1550 alsoincludes two segments, 1560 and 1565, in which the value of Fouriertransform 1525 is not zero. As a result of the contributions of segments1560 and 1565 to this integral along line 1550, the value of the Fouriertransform of slice 1540 at point P 1545 will not be zero. Thus, slice1540 is not bandlimited.

Embodiments of the invention exploit the orthogonality property of thewavelet noise to create bandlimited slices of noise data from higherdimensional bandlimited noise sets. An embodiment of the inventiondefines the value of noise data in a slice of A dimensions as theprojection of the noise set of B dimensions on to the basis functions ofA dimensions, where B is greater than A. For example, in a planar slicealigned with an axis of the noise data set, such that the slice has aplane equation 2^(j)z=z₀, the value of noise data at a point on theplanar slice is defined asNoise_(z0)(2^(j) x,2^(j) y):=∫Noise_(N)(2^(j) x,2^(j) y,2^(j) z)φ(z−z₀)dz.

The projection of noise data from a bandlimited noise set onto the basisfunctions of a lower dimensional slice is equivalent to performing a(B-A) dimensional area integral orthogonal to the slice, where theintegrand is the noise data weighted by a scaling function, φ(z),centered at the point of intersection. In another embodiment, the (B-A)dimensional area can be integrated as a sequence of line integrals orother lower dimensional area integrations. The scaling function φ(z) canbe a piecewise constant scaling functions, such as Haar basis functions;piecewise polynomial scaling functions, such as uniform B-spline basisfunctions of any degree; bandlimited scaling functions, such as sincfunctions; and Daubeschies scaling functions.

This definition of the value of a point of a slice of a bandlimitednoise function guarantees that the slice will be orthogonal to lowerresolution versions of itself. As a result, the slice of bandlimitednoise data will be also be bandlimited.

In a further embodiment, the value of the integral used for projectionof noise data onto a lower dimensional slice can be approximated usingthe quadrature techniques discussed above. In this application, theintegral of the bandlimited noise function times the scaling functionφ(z) can be approximated as the sum of the bandlimited noise functiontimes the convolution of the noise function's scaling basis functionwith itself. In the example of bandlimited noise function using aquadratic B-spline scaling basis function, this convolution results in aquintic function. In yet a further embodiment, the quintic function canbe approximated as a widened version of the quadratic B-spline scalingbasis function, similar to that described in FIGS. 14A and 14B. In thisembodiment, the integral is approximated using a widened quadraticB-spline basis function as the filter kernel in the normal direction ofthe slice. In an embodiment, the widened quadratic B-spline basisfunction has twice the width of the unwidened version

In an embodiment, the filter kernel in the tangent direction of theslice is unchanged in form and characteristics used to evaluate thebandlimited noise function. For example, the renderer's filter kernelcan be used to evaluate the noise function in the tangent direction. Inanother example, the approximation

$\sum\limits_{q = 1}^{Q}{B_{q}{\overset{\sim}{N}\left( x_{q} \right)}}$discussed above can be used to evaluate the noise function in thetangent direction and compensate for the renderer's low sampling rate.In still another example, if Ñ(x_(q)) includes a quintic function,resulting from the use of quadratic B-splines for the renderer filterkernel and the noise function scaling basis functions, then this quinticcan be further approximated as a widened version of the quadraticB-spline basis function.

FIG. 16 illustrates a bandlimited noise set 1600 including discretenoise coefficients and a slice 1605 to be extracted. The value of thenoise slice 1605 at point P 1610 is determined using a projection of thenoise data onto the basis functions of the lower-dimensional slice, asdescribed above. In this example, the value of this projection at pointP 1610 is determined from the linear combination of noise coefficientsnear point P 1610 that are weighted by the value of filter kernel 1620at the location of each noise coefficient. In this example, the filterkernel 1620 is composed of a quadratic B-spline basis function of usualwidth in the tangent direction 1625 and a quadratic B-spline basisfunction of double width in the normal direction 1630. At the locationof each noise coefficient, such as noise coefficient 1635, interpolationof the filter kernel is used to define a weight to be applied to thenoise coefficient value.

Because of the orthogonality properties of wavelet basis functions, suchas the Haar basis or the quadratic B-spline basis, the contribution ofnoise coefficients outside of the filter kernel 1620 are zero ornegligible. The widened basis function in the normal directionintegrates noise data at the same frequency scale at which thebandlimited noise disappears, due to its bandlimited nature.

The projection of noise data onto a slice is axis-aligned. A commonapplication of this axis-aligned projection is the projection of a fourdimensional bandlimited noise set (indexed by x,y,z, and time) onto athree dimensional slice (x,y,z). In this application, the projection isaligned with the time axis. In other applications, such as the aboveexample, the projection may not be axis-aligned. For non-axis-alignedprojections and non-integer resolution levels, the deviation fromorthogonality is often negligible.

In these applications, the normal and tangent vectors of a point in aslice of noise data may be derived from derivatives of an associatedsurface. In other applications, normal and tangent vectors can beassigned to a point in a slice of noise data using other techniques,such as when the point is not associated with a differentiable surface.

It should be noted that the above described noise functions can beincorporated into any rendering technique, for example ray-tracing orscanline rendering, to create one or more images or frames using noisefunctions in combination with models, lighting, shading, texturemapping, and any other image processing information. Additionally, theabove described noise functions can be created and utilized in functionsand programs implemented in any type of general purpose programminglanguage, such as C, as well as rendering and shading specificlanguages, such as the Renderman shading language.

Further embodiments can be envisioned to one of ordinary skill in theart after reading the attached documents. In other embodiments,combinations or sub-combinations of the above disclosed invention can beadvantageously made. The block diagrams of the architecture and flowcharts are grouped for ease of understanding. However it should beunderstood that combinations of blocks, additions of new blocks,re-arrangement of blocks, and the like are contemplated in alternativeembodiments of the present invention.

The specification and drawings are, accordingly, to be regarded in anillustrative rather than a restrictive sense. It will, however, beevident that various modifications and changes may be made thereuntowithout departing from the broader spirit and scope of the invention asset forth in the claims.

1. A method of forming a computer generated image that includescharacteristic information of pixels, wherein the characteristicinformation of at least a portion of the pixels is determined at leastin part by sampling noise data defined by a scene including athree-dimensional model of an object, the method comprising: creatingfirst noise data using a processor; constraining the first noise datausing a processor such that the first noise data has a frequencyspectrum contained substantially between first and second frequencyvalues; wherein a sum of the frequency spectrum of the first noise dataand a copy of the frequency spectrum of the first noise data issubstantially flat, wherein the copy of the frequency spectrum istranslated in frequency by the difference between the first and secondfrequency values wherein the frequency spectrum of the first noise datadiminishes abruptly outside of a range between the first and secondfrequency values, wherein the frequency spectrum of the first noise datadiminishes from a significant magnitude to approximately zero within apower of two from the first and second frequency values.
 2. The methodof claim 1, wherein the first noise data is comprised of a linearcombination of a first set of locally supported scaling basis functions.3. The method of claim 1, wherein the first noise data lacks adiscernable pattern.
 4. The method claim 3, wherein the first noise datais randomly generated.
 5. The method of claim 3, wherein the first noisedata is pseudo-randomly generated.
 6. The method of claim 1, furthercomprising: creating second noise data having a frequency spectrumcontained substantially between third and fourth frequency values; andcombining the first noise data and the second noise data.
 7. The methodof claim 6, wherein creating the second noise data comprises: scalingthe first noise data to a first resolution to form a scaled noise data,wherein the first resolution corresponds to the third frequency; andsetting the second noise data to the scaled noise data.
 8. The method ofclaim 6, wherein creating the second noise data comprises: creatingsecond noise data; and constraining the second noise data such that afrequency domain transform of the second noise data produces a frequencyspectrum contained substantially between the third and the fourthfrequency values.
 9. The method of claim 6, wherein the first noise datais comprised of a linear combination of a first set of scaling basisfunctions defining a first vector space and the second noise data iscomprised of a linear combination of a second set of scaling basisfunctions defining a second vector space, wherein the second vectorspace is included in the first vector space.
 10. The method of claim 9,wherein the first set of scaling basis functions is orthogonal to thesecond set of scaling basis functions, such that such that every vectorin the second vector space can be represented in the first vector spaceand at least one vector in the first vector space cannot be representedin the second vector space.
 11. The method of claim 1, wherein thesecond frequency value is an integer multiple of the first frequencyvalue.
 12. The method of claim 11, wherein the integer multiple is two.13. A method of forming a computer generated image that includescharacteristic information of pixels, the method comprising: creatingnoise data using a processor; using a processor, specifying a filterkernel adapted to sample scene data including a contribution of thenoise data to determine the characteristic information of at least aportion of the pixels of the computer generated image; and constrainingthe noise data such the contribution of the noise data diminishessubstantially at an aliasing frequency, wherein the filter kernel isadapted to approximate a renderer filter kernel, wherein the filterkernel is based on a convolution of the renderer filter kernel and ascaling basis function of the noise data, and wherein the filter kernelis a widened version of the scaling basis function adapted toapproximate the convolution.
 14. The method of claim 13, wherein thefilter kernel is adapted to be used by a renderer.
 15. The method ofclaim 13, wherein the noise data is comprised of a linear combination oflocally supported scaling basis functions.
 16. The method of claim 13,further comprising: constraining the noise data such that thecontribution of the noise data diminishes substantially at a minimumfrequency.
 17. The method of claim 16, wherein the aliasing frequency isan integer multiple of the minimum frequency.
 18. The method of claim17, wherein the integer multiple is two.